Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-2q^2 + 32q - 128}{-9q^2 + 45q + 216}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-2(q^2 - 16q + 64)} {-9(q^2 - 5q - 24)} $ $ t = \dfrac{2}{9} \cdot \dfrac{q^2 - 16q + 64}{q^2 - 5q - 24} $ Next factor the numerator and denominator. $ t = \dfrac{2}{9} \cdot \dfrac{(q - 8)(q - 8)}{(q - 8)(q + 3)}$ Assuming $q \neq 8$ , we can cancel the $q - 8$ $ t = \dfrac{2}{9} \cdot \dfrac{q - 8}{q + 3}$ Therefore: $ t = \dfrac{ 2(q - 8)}{ 9(q + 3)}$, $q \neq 8$